FSMQ Level 3 (legacy) Working with algebraic and graphical techniques scheme of work
This FSMQ requires a total of 60 guided learning hours that could be used in a variety of ways, such as 2 hours per week for 30 weeks, 4 hours per week for 15 weeks, or 5 hours per week for 12 weeks. Before starting this course learners should be able to:
 plot by hand accurate graphs of paired variable data and linear and simple quadratic functions (including the type y = ax^{2} + bx + c) in all 4 quadrants
 recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions (including the type y = kx^{2} + c)
 fit linear functions to model data (using gradient and intercept)
 rearrange basic algebraic expressions by collecting like terms, expanding brackets and extracting common factors
 solve basic equations by exact methods including pairs of linear simultaneous equations
 use power notation (including positive and negative integers and fractions)
 solve quadratic equations by factorising and using the formula (must be memorised).
A suggested work scheme for this unit is given below. It includes some revision of the above as well as the other topics and methods to be covered, but note that you will also need to allow time for students to complete an AQA Coursework Portfolio. The Coursework Portfolio requirements are listed on page 56 of the AQA Advanced FSMQ specification and also on page 101 of the AS Use of Mathematics specification. The assignments below provide some examples of the sort of work your students could include in their portfolios, but if possible you should use work that is more relevant to their other studies or interests..
The following techniques should be introduced as soon as possible and used throughout the course:
 using a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
 doing calculations without a calculator using written methods and mental techniques
 graph plotting using computer software or a graphical calculator, and using trace and zoom facilities to find significant features such as turning points and points of intersection
 checking calculations using estimation, inverse operations and different methods.
Topic area 
Content 
Nuffield resources 
Linear functions (3 hours) 
Revise the main features of graphs of direct proportional (y = mx) and linear (y = mx + c) functions. Fit such functions to real data using gradients and intercepts. Understand whether it is appropriate or not to use a particular function to model data by consideration of intercepts, long term behaviour (etc.) in real world terms. Use error bounds to consider a range of possible functions to model data. Solve linear simultaneous equations using graphical and algebraic methods. 
Linear graphs 
Graphs of functions in Excel 

Interactive graphs 

Graphic calculators 

Using the CASIO fx7400G PLUS 

Graphic calculator equations 

Car bonnet 

Match linear functions and graphs 

Simultaneous equations on a graphic calculator 
Topic area 
Content 
Nuffield resources 
Quadratic functions 
Draw graphs of quadratic functions of the form: · y = ax^{2} + bx + c · y = (rx – s )(x – t) · y = m(x + n)^{2} + prelating the shape, orientation and position of the graph to the constants and relating zeros of the function f(x) to roots of the equation f(x) = 0. Fit quadratic functions to real data. Revise solving quadratic equations by: · factorising · using the formula
Rearrange any quadratic function into the forms Find maximum and minimum points of quadratics by completing the square. 
Test run 
Model the path of a golf ball 

Broadband A, B, C Presentation shows the algebra version. 

Two on a line and three on a parabola 

Factor cards 

Water flow 

Completing the square 

Gradients of curves, maxima and minima (3 hours)

Calculate and understand gradient at a point on a graph using tangents drawn by hand (and also using zoom and trace facilities on a graphic calculator or computer if possible). Use and understand the correct units for rates of change. Interpret and understand gradients in terms of their physical significance. Identify trends of changing gradients and their significance both for known functions and curves drawn to fit data. 
Tin can 
Maximum and minimum problems 
Topic area 
Content 
Nuffield resources 
Power functions and inverse functions (3 hours) 
Draw graphs of functions of powers of including where is a positive integer, , , and Find the graph of an inverse function using reflection in the line y = x. Solve polynomial equations of the form ax^{n} = b 
Interactive graphs 
Growth and decay (8 hours) 
Draw graphs of exponential functions of the form and (m positive or negative) and understand ideas of growth and decay. Draw graphs of natural logarithmic functions of the form y = a ln(bx) and understand the logarithmic function as the inverse of the exponential function. Solve exponential equations of the form Learn and use the laws of logarithms · · · to convert equations involving powers to logarithmic form and solve them (using both base 10 and natural logarithms). 
Growth and decay 
Population growth 

Calculator table 

Ozone hole 

Climate prediction A and B 

Cup of coffee 
Topic area 
Content 
Nuffield resources 
Transformations of graphs (4 hours) 
Use: · translation of y = f(x) parallel to the x axis to give y = f(x + a) · stretch of y = f(x) parallel to the y axis to give y = af(x) · stretch of y = f(x) parallel to the x axis to give y = f(ax)

Sea defence wall (assignment) 
Trigonometric Functions (8 hours) 
Draw graphs of · · Learn the general shape and position of trigonometric functions and use the terms amplitude, frequency, wavelength, period and phase shift correctly. Fit trigonometric functions to real data. Solve trigonometric equations of the form and 
Coughs and sneezes 
SARS A and B (assignments) 

Sunrise and sunset times (assignment) 

Tides (assignment) 
Topic area 
Content 
Nuffield resources 
Linearising data (6 hours) 
Determine parameters of nonlinear laws by plotting appropriate linear graphs, for example:
·
·
·
·
· and 
Log graphs  earthquakes 
Gas guzzlers 

Smoke strata 

Earthquakes (ln version & log version) 

Revision (6 hours) 


Page last updated on 25 January 2012